On the boomerang spectrum of functions located at a distance of at most two from the class APN functions

The boomerang attack, proposed in 1999, which is a variant of the difference attack, is interesting because it can reduce the reliability of the cipher despite its good differential properties. The parameter that characterises the resistance of a cipher to this method has been named boomerang uniformity. We consider boomerang characteristics of functions at distance one from the class of APN functions and permutations at distance two from the class of APN permutations. Boomerang spectra of values of such sets with conditions of reachability of these values are found. Upper bounds for the values of boomerang uniformity are given.